Continuous dependence inequalities for a class of quasilinear parabolic problems (Q2505203)

Let \(u(x,t)\) be the classical solution to the following initial-boundary value problem \[ \begin{cases} \Delta u -{{\partial u}\over{\partial t}}=-f(u), & x\in \Omega,\;\;t\in(0,T),\\ u(x,t)=0, & x\in \partial\Omega,\;t\in [0,T],\\ u(x,0)=g(x), & x\in \Omega;\;\;g(x)=0,\;x\in\partial\Omega. \end{cases} \] The authors obtain continuous dependence inequalities for \(u(x,t)\) and \(| \nabla_x u(x,t)| \) when the data \(f\) and \(g\) are subject to variations.